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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
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Capitolo 3, Problema 55f

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sin t - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.
The acceleration of the oscillator is a(t) = v′(t). Find and graph the acceleration function.

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To find the acceleration function, we first need to determine the velocity function v(t). The velocity is the derivative of the position function y(t) with respect to time t. So, we start by differentiating y(t) = 30(sin t - 1).
The derivative of y(t) = 30(sin t - 1) with respect to t is v(t) = 30 * cos(t). This is because the derivative of sin(t) is cos(t), and the constant -1 becomes 0 when differentiated.
Now, to find the acceleration function a(t), we need to differentiate the velocity function v(t) = 30 * cos(t) with respect to t.
The derivative of v(t) = 30 * cos(t) is a(t) = -30 * sin(t). This is because the derivative of cos(t) is -sin(t).
To graph the acceleration function a(t) = -30 * sin(t), note that it is a sinusoidal function with amplitude 30, period 2π, and it oscillates between -30 and 30. The graph will be a sine wave starting at 0 when t = 0, going downwards initially because of the negative sign.

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Concetti chiave

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Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of a function with respect to its variable. In this context, we need to differentiate the position function y(t) to find the velocity v(t) and then differentiate again to find the acceleration a(t).
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Finding Differentials

Acceleration

Acceleration is defined as the rate of change of velocity with respect to time. In this problem, the acceleration function a(t) is derived from the velocity function v(t), which is itself obtained by differentiating the position function y(t). Understanding how to compute and interpret acceleration is crucial for analyzing the motion of the oscillating object.
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Derivatives Applied To Acceleration

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate system to visualize its behavior. For the acceleration function a(t), we will need to calculate its values over a range of t and then plot these points to observe how the acceleration changes over time. This visual representation helps in understanding the dynamics of the oscillating object.
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Graph of Sine and Cosine Function
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